Dynamics of the vortex-particle complexes bound to the free surface of superfluid helium
DDynamics of the vortex-particle complexes bound to the free surface of superfluidhelium
P. Moroshkin , , ∗ P. Leiderer , K. Kono , , S. Inui , and M. Tsubota RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, 351-0198 Saitama, Japan Okinawa Institute of Science and Technology, 1919-1 Tancha, Onna-son, 904-0495 Okinawa, Japan Department of Physics, University of Konstanz,Universit¨atstrasse 10, 78464 Konstanz, Germany Department of Electrophysics, National Chiao Tung University, Hsinchu 300, Taiwan and Department of Physics, Osaka City University, 3-3-138 Sugimoto, 558-8585 Osaka, Japan (Dated: October 3, 2018)We present an experimental and theoretical study of the 2D dynamics of electrically chargednanoparticles trapped under a free surface of superfluid helium in a static vertical electric field. Wefocus on the dynamics of particles driven by the interaction with quantized vortices terminating atthe free surface. We identify two types of particle trajectories and the associated vortex structures:vertical linear vortices pinned at the bottom of the container and half-ring vortices travelling alongthe free surface of the liquid.
Quantized vortices are topological defects that existin superconductors and in superfluids, such as superfluidhelium, atomic Bose-Einstein condensates and exciton-polariton condensates. Vortices in superfluids can becreated by microscopic impurity particles [1] and by vari-ous mesoscopic objects [2–4] moving faster than a certaincritical speed. Suspended microparticles interact withvortices and become bound to them [5–7]. Free motionof these particle-vortex complexes has been observed inbulk superfluid He [8–11]. Related effects have also beenstudied in superfluid He nanodroplets containing impu-rity atoms and nanoparticles [12].Dynamics of a free surface is another important topicin the research on quantum fluids. The free surfaces ofsuperfluid He and He have been investigated using freeelectrons and He + ions as probes. Both types of chargedparticles can be localized at the free surface and drivenparallel to it by the external electric field. This approachhas led to the observations of anomalous Hall effect oftopological origin in superfluid He-A [13, 14], of Majo-rana surface states in superfluid He-B [15, 16].The research on quantum hydrodynamics has a longhistory [17], however most efforts have been devoted tobulk. Until recently, little attention has been paid tothe phenomena involving quantized vortices near solidboundaries or surfaces [18], in particular a free sur-face. Here we present a new experimental and theoreticalstudy of the motion of particle-vortex complexes in su-perfluid He. Our experiments visualize the motion of thetip of a vortex terminating at a free surface. The vor-tex is bound to an electrically charged particle which istrapped under the free surface due to the applied elec-tric field and the surface tension. Our observations com-bined with the numerical modeling allow us to identifytwo types of vortices that form particle-vortex complexesunder these conditions: a linear vortex stretched in a ∗ [email protected] pulsedlaser U bottom cw laser U back U front target θ camera liquid He FIG. 1: Left - experimental sample cell; right - typical imageof trapped particles (bottom view). vertical direction and a half-ring vortex with both endsterminating at the free surface.Our experiments are performed in a helium-bath cryo-stat with optical access via four side-windows and a win-dow in the bottom. The temperature is adjusted in therange of T = 1.35 - 2.17 K by pumping on the liquidHe in the bath. The sample cell is immersed in the Hebath and is filled up to a certain height by condensing Hegas from a high pressure gas cylinder. A system of threehorizontal flat electrodes shown in Fig. 1(a) is used tocreate a static electric field, with a predominantly verti-cal orientation. The bottom electrode is transparent andthus allows us to monitor the interior of the cell and inparticular the liquid He surface via the window at thebottom.The tracer particles are produced inside the cell andare trapped at the free surface by the technique devel-oped in our recent studies [19–21]. A frequency-tripledpulsed diode-pumped solid-state (DPSS) laser ( λ = 355nm) with a pulse energy of 70 µ J is used to ablate ametal (Cu, Ba, Dy) target submerged in liquid He. Asingle laser pulse is sufficient to produce several tens orhundreds of metallic micro- and nanoparticles which be-come electrically charged either in the ablation plume, orby touching the bottom electrode. The charged particles a r X i v : . [ c ond - m a t . o t h e r] S e p then rise up and become trapped under the free surface ofthe liquid He. The trapping potential is provided by theelectric field that pulls the particles upwards and by thesurface tension of liquid He which does not allow themto cross the liquid-gas interface.In order to visualize the injected particles, we use scat-tered light from the beam of a cw frequency-doubleddiode laser with a wavelength λ = 480 nm. The laserbeam is expanded in a horizontal direction and illumi-nates the particles via a side window. The motion ofparticles along the surface is recorded by a digital videocamera, installed under the bottom window of the cryo-stat and operated at a frame rate of 100–5300 fps. Theangle θ between the cw laser beam and the free surfaceof the liquid is adjusted in the range of ± ◦ in orderto get rid of dark zones appearing due to surface waveswhich are excited by slight vibrations of the experimentalset-up.The size of each individual particle is not known. Thescanning electron microscope (SEM) study [19] of thedeposites collected after the experiment has revealed twotypes of particles: spheres with diameters in the range of20–500 nm and nanowires with a uniform diameter of ≈ i.e. horizontal) is equal to zero.Surface waves tilt the free surface locally, thus produc-ing at the particle location a non-zero component of theelectric field parallel to the surface. This leads to collec-tive particle oscillations in the XY plane that is nicelyresolved by our technique. All particles within the samearray oscillate in phase, with a coherence time of severalseconds and with an amplitude of 50 µ m, or less. Underquiet conditions, the oscillations become barely visible.In Figs. 2(a) and (b) we show the maps of particle tra-jectories from two different video recordings, obtained bytracing the motion of all particles in each recording withthe help of Diatrack [22] software. All regular particlesare represented as well separated blobs, their size reflect-ing the amplitudes of the particle oscillations along X and Y directions.In addition to the collective oscillations induced by thesurface waves, we observe also a small number of anoma-lous particles whose motion is strikingly different fromthat described above. Two anomalous trajectories can (a) (b) Y ( mm ) X (mm)4.8 5.0 5.2 (c) (d) Y ( mm ) X (mm) 6.54.5 5.04.05.35.1 4.9 5.1 7.07.58.08.53.0 3.5
FIG. 2: (a), (b) Maps of particle trajectories. (c), (d) Tracedtrajectories of two anomalous particles from (a). Experimen-tal conditions in (a), (c), (d): T = 1.35 K, U front = +100 V, U bottom = -110 V, cw laser power 42 mW; in (b): T = 2.1 K, U front = -116 V, U bottom = 0, cw laser power 35 mW. be seen in Fig. 2(a) and one in Fig. 2(b). We distin-guish two types of anomalous particles. The particles oftype 1 oscillate with a larger amplitude and out of phasewith the normal particles. Their trajectory representsa sequence of circular loops 50–200 µ m in diameter, asshown in Fig. 2(c). The particles of the second typemove more or less straight at approximately constant ve-locities, sometimes across the entire field of view of thecamera. The trajectory of such a particle typically con-sists of straight or slightly curved segments with sharpturning points, as can be seen in Figs. 2(a), (b) and (d).In many cases this motion is quasiperiodic. At the turn-ing points, the particle abruptly changes the direction ofmotion. In some cases, the velocities on different seg-ments are different, but are well reproduced over manyperiods. Both types of anomalous motion persist for thewhole duration of the observation of the particular setof particles, sometimes up to several minutes. X ( t ) and Y ( t ) curves typical for the type 1 and type 2 particlesare shown in Figs. 3 (a) and (b), respectively.When an anomalous particle of type 2 moves throughthe cloud of trapped normal particles, it experiences nu-merous collisions. The turning points are usually asso-ciated with some of those collisions. In many other col-lisions, the anomalous particle keeps moving along thestraight trajectory almost without any deviation. In-stead, the normal particles are pushed aside, make smallloops and return to their equilibrium positions behindthe passed anomalous particle. Several such events canbe seen along the straight track of the anomalous particle X , Y ( mm ) X(t) Y(t)(a)
Time (s) X , Y ( mm ) X(t)Y(t)(b)
FIG. 3: X ( t ) and Y ( t ) coordinates of anomalous particles:(a) - type 1, T = 1.35 K, U top = +100 V, U bottom = -110 V,cw laser power 42 mW; (b) - type 2, T = 1.35 K, U top = -250V, U bottom = +300 V, vertical arrow marks the moment whenthe cw laser power is changed from 45 to 235 mW. in Fig. 2(a).The power of the cw blue laser illuminating the trappedparticles can influence the particle motion along the sur-face. Higher laser power leads to a more active particlemotion, lager amplitude of particle oscillations (see Fig.3(b)) and a larger number of anomalous particles movingat higher velocities.Particle motion in our experiment can be induced bysurface waves and by a counterflow induced by the heat-ing of the cell walls and particles themselves by the cwlaser illumination. However, both mechanisms result ina collective motion of all particles and can not explainthe individual anomalous motion of one or several par-ticles within a large array. It is expected that somenumber of quantized vortices is always present in super-fluid He under the conditions of our experiments. It hasbeen demonstrated both theoretically [5–7] and experi-mentally [8, 9, 11] that solid nano- and microparticlesbecome trapped by the vortices and move together withthem. We therefore suggest that the anomalous particlesdiffer from the regular ones due to their binding to quan-tized vortices and their motion is driven by the dynamicsof the vortex.Our theoretical model of the coupled dynamics ofquantized vortices and particles is based on the vortexfilament model [23]. The simplest configuration shownin Fig. 4(a) consists of a single vertical straight vortexfilament and a particle attached together. It is assumedthat the upper end of the filament is connected at the cen-ter of the spherical particle and both move at the samespeed. The motion is excited by a collision between thestraight filament at rest with a small vortex ring.The trajectory of the particle at the surface followingthe excitation is shown in Fig. 4(c). The collision leads to a transient dynamics, when the particle moves alonga spiral trajectory away from the equilibrium position.This spiral motion decays and is replaced by a smallerscale circular motion that does not depend on the detailsof the excitation and therefore is intrinsic for this sys-tem. The mechanism can be summarized as follows. Themain part of the long vortex filament remains straightand vertical. It creates a concentric velocity field v s,BS that drops inversely proportional to the distance fromthe center. Small displacements of the particle togetherwith the upper segment of the filament do not perturbthis velocity profile significantly. The total superfluid ve-locity v s is expanded into two terms: v s = v s,BS + v s,LIA ,where v s,LIA is obtained in a local induction approxima-tion and is determined by a local curvature of the fila-ment s (cid:48)(cid:48) . When the particle is kicked out off axis of thevortex velocity field, it shows the circular motion abouta new equilibrium location which is determined by bal-ancing the two forces acting on the vortex segment: theMagnus force F M ∝ s (cid:48) × ( ˙ s − v s,BS ) and the tensionforce F t ∝ s (cid:48)(cid:48) . Here, s is the filament and the prime andthe dot symbols represent the derivatives with respectto arc length and time, respectively. The Magnus forceacts outward and is dominant when v s,BS is large. Onthe other hand, the tension only depends on the localcurvature of the filament. As the particle gets fartherthe local curvature of the vortex becomes larger and thetension force that pulls the particle backward becomesdominant. The circular motion strongly resembles theexperimentally observed anomalous dynamics of type 1(see Fig. 2(c)), although the computed trajectory radiiare significantly smaller than experimental. As discussedin [24], the radius increases with the size of the parti-cle. In the experiment, the particle heated by the laserlight may become surrounded by a shell of normal fluidHe which may lead to the increased effective particle sizeand a larger trajectory radius.In order to explain the anomalous particle motion oftype 2, we have to consider more complex systems in-cluding several quantized vortices. In particular, a pairof linear vortices orthogonal to the free surface and ro-tating in two opposite directions is expected to moveparallel to the surface at a constant speed that is in-versely proportional to the distance between the vor-tices d : v d = Γ / πd . Γ here is the quantum of thecirculation: Γ = h/M He = 9 . × − cm /s. Suchcomplexes, also known as quantized vortex dipoles, havebeen predicted and investigated theoretically [25–27].More recently, these interesting macroscopic quantumobjects were observed in atomic Bose-Einstein conden-sates (BEC) [4, 28–30] and in exciton-polariton conden-sates [31, 32]. Vortex dipoles in superfluid He have beenpredicted theoretically [33] and the interaction and trap-ping of particles by vortex dipoles and larger clusters ofquantized vortices was modeled numerically in [7]. How-ever, no experimental observations have been reportedup to date.Our calculations demonstrate that two counter- (a) (b)(c) (d) FIG. 4: Schematic drawings of tracer particles trapped byquantized vortices: (a) linear vortex, (b) half-ring vortex. (c)Calculated particle trajectory corresponding to (a). (d) Cal-culated trajectories of two half-ring vortices created as a resultof multiple reconnections of three linear vortices. rotating linear vortices at a close distance become un-stable. The two filaments approach each other and re-connect in one or several places producing several vortexrings of different sizes. Remarkably, the upper segmentsof the two filaments touching the free surface upon thereconnection transform into a surface-bound vortex of anew type: a half-ring with a radius R equal to one halfof the distance between the two parent vortices and withboth ends terminated at a free surface. Similar to the mo-tion of vortex dipoles and vortex rings in the bulk [34],the half-ring vortex moves along the direction of the ringaxis, with a velocity v hr = Γ / πR . With the both endsterminated at the free surface, the half-ring maintainsa vertical orientation and moves parallel to the surface.This translational motion is accompanied by the oscilla-tions of the half-ring radius, that depend on the details ofthe collision and reconnection of the two initial vortices.Fig. 4(d) shows the motion of two half-ring vortices cre-ated as a result of the reconnections of three linear vor-tices. The trajectory of each half-ring is represented bythe traces of its two tips along the surface.The scenario outlined above allows us to assign theanomalous particle motion of type 2 to the particlestrapped by half-ring vortices traveling along the free sur-face of superfluid He. The particle of a sub-micron diam-eter is supposed to be trapped at one end of the vortex,as shown in Fig. 4(b). Experimentally measured particlevelocities fall in the range of 0.1–10 cm/s, which corre-sponds to the radii of the half-ring vortices of 0.1–10 µ m.The nearly constant velocity of many type 2 particles im-plies that the radius of the half-ring in each case remainsconstant over a long period of time. When the particleattached to one end of the half-ring hits an obstacle such as another charged particle or the inhomogeneity of theexternal electric field at the edge of the capacitor, it slowsdown or stops abruptly and the other end of the half-ringmoves around it. As a result, the half-ring changes itsorientation and starts moving away from the obstacle.Similar effect can be produced by collisions of the com-plex with other co-rotating vortices. The collisions mayalso lead to the abrupt change of the half-ring radius andthe corresponding velocity modulus. The particle-vortexinteraction is sufficiently strong that the particle drivenby the vortex is able to overcome the Coulomb repulsionand approach close to other particles and even kick themout from their equilibrium positions in the trap.Our observations clearly demonstrate that upon an in-crease of the illuminating light intensity some regularparticles start moving as anomalous of type 2. We sug-gest that the heating of individual particles by the intenselaser light induces a strongly localized radial counterflowaround the particle which exceeds the critical velocityand leads to the creation of new vortices pinned at theparticle. For some particles, at a high laser power we ob-serve frequent and chaotic changes of the direction andmagnitude of their velocity which may be a signature ofmultiple vortices interacting with the particle.Even more intriguing is the behavior of the particlesthat upon the increase of the light intensity increase theirspeed, but retain their highly regular periodic characterof motion, as shown in Fig. 3 (b). The heating of theparticle thus results in the shrinking of the half-ring vor-tex bound to it. The effect can be attributed to theheat-induced counterflow around the particle, in whichthe superfluid component is moving towards the particleand the normal component - away from it. The free endof the superfluid half-ring vortex is thus pushed towardsthe particle. It is thus possible to manipulate the size ofthe half-ring vortices and control their motion by shin-ing the laser light onto the trapped particle, which opensnew perspectives for experiments.In summary, we have developed a new method for vi-sualizing the motion of quantized vortices terminating ata free surface of superfluid He that relies on the usage ofelectrically charged tracer particles trapped at the sur-face. We interpret our observations in terms of two typesof surface-bound vortices with the characteristic dynam-ics: a linear vertical filament and a half-ring. The formerremains pinned at the bottom, with the upper end andthe attached particle moving along a circular trajectory ≈ µ m in diameter. The latter moves together with thetrapped particle along a straight line and turns aroundabruptly when meets an obstacle. The vortices interactwith the laser light via the heating of the trapped particleand the resulting local counterflow. Acknowledgments